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Fig. 5.3 shows the evolution of the mass of condensed baryons (stars and cold gas,

M∗+Mcold) as a fraction of the total baryon mass as a function of redshift for the three

mass bins. There is fairly good agreement between the simulations and the NF model over the whole redshift range, although the SAM is a little low at high redshift and a bit high at low redshift, particularly in the high and intermediate mass bin. In the SN model, the condensed baryon fraction is lowered by an almost fixed factor relative to the NF model, and is significantly lower than the simulation results. This suggests that the “thermal feedback” implemented in the simulation is less effective than that included in the SAM. It may seem curious that the SNWM model results are higher

than the SN model, in fact close to the NF model in the high and intermediate mass bin. This is because the SNWM includes metal cooling, while the SN model does not. The enhanced cooling rates partly compensate for the removal of cold gas via the SN- driven winds. Finally, one can see that in the FULL model, the AGN feedback begins to quench star formation in the massive halos after about z _∼ 2, while it has little

effect on the lower mass bins.

In Fig.5.4the evolution of the mean cold gas fraction of the central galaxy is plotted (note that in Fig.5.3 cold gas plus stars is plotted). The efficiency of the conversion of cold gas to stars is clearly very different between the simulations and SAMs. In both cases (simulations and SAMs), the final cold gas fraction is increasing with decreasing halo mass. For the NF model, the SN and the SNWM model the cold gas fraction varies only slightly over time and is significantly higher (about an order of magnitude since z = 2) than for the simulations. This shows that the inclusion of SN feedback

has little impact on the gas fractions of galaxies. Only the FULL model shows a much stronger decrease of the gas fraction with cosmic time for massive galaxies due to the radio mode feedback. The initial cold gas fraction, at high redshifts 4 < z < 8, is

almost the same for the simulations and SAMs. With evolving cosmic time the cold gas content decreases more rapidly in the simulations due to the more efficient conver- sion into stars. The cold gas in the simulations is already converted into stars at high redshift and there is almost no more cold gas left to turn into stars at lower redshifts. This is similar to the results found in the comparison of Cattaneo et al. (2007).

Figure 5.3: The evolution of the fraction of the condensed baryons M∗ +Mcold of

the total baryon mass as a function of redshift for the three mass bins. The condensed baryon fraction of all simulations is in reasonable agreement with the NF model at all redshifts. High-mass and intermediate-mass simulations agree well with the SNWM model. The effect of AGN feedback in the FULL model can be clearly seen in the high-mass bin. In the low-mass bin the SN wind feedback (SNWM) makes the biggest difference.

Figure 5.4: Evolution of mean cold gas fraction Mcold/(fbar ×Mhalo) of the central

galaxies. Mass bins and colors are the same as in Fig. 5.2. In all simulations the cold gas is depleted more efficiently than in the SAMs due to the large star formation efficiency at high redshifts. The red dotted line shows the cold gas fraction assuming ten times higher efficiency for star formation in the NF model (see Eq. 5.5).

Figure 5.5: Comparison of the stellar baryon fraction of the central galaxy between simulations and SAMs. The mass bins and colors are the same as in Fig. 5.2. For all mass bins the simulations agree best with the NF SAM, but forming significantly more stars already higher redshifts z > 1. This discrepancy can be accounted for if the star formation efficiency parameter in the SAMs is increased by a factor 10 for the NF case (red dashed lines). At high masses the AGN feedback (FULL) and at the low masses the SN feedback (SN and SNWM) reduce the stellar baryon fractions in the SAMs.

into stars in the central galaxy M∗/(f_bar ×M_halo), sometimes termed “baryon conver-

sion efficiency” (Guo & White, 2009; Moster et al., 2010). In general, all simulations predict a decreasing (high-mass) or almost constant (low-mass) conversion efficiency with redshift, whereas most SAMs predict increasing conversion efficiencies with the exception of high-mass galaxies in the FULL model with AGN feedback.

At low redshiftz <0.6the conversion efficiencies agree well between the simulations

and the NF model, with higher values for lower mass galaxies. However, at high redshifts z > 1 the conversion efficiencies are significantly higher for the simulations.

This is in contrast to the results ofCattaneo et al.(2007), where the stellar masses agree at high redshift, but the SAM masses are larger than in simulations at low redshift. The difference in the behavior of the SAMs and the simulations can be explained in terms of star formation efficiency. We changed the normalization of the SK-relation in the SAM by introducing a factor τ∗ in Eq. 4.4:

ΣSFR=

AKS

τ∗ ΣNK

gas, (5.5)

with τ∗ ≈0.1. The results of the NF model with this elevated star formation efficiency

for the stellar and cold gas mass evolution is shown in Figs. 5.4 and 5.5. At high redshift, gas is more efficiently depleted and converted into stars, resulting in a better agreement between this ‘high SFE’ model and the simulations. However, for the high- mass and intermediate-mass bin, the ‘high SFE’ model over-predicts the stellar fraction at low redshifts, suggesting that the Cattaneo et al.(2007) SAMs may also have had a higher SFE, and this could explain the discrepancy between their results and the initial results presented here. In all three mass bins, the NF model produces the most massive stellar components. Again, the more efficient cooling due to metals in the SNWM and the FULL model is cancelled by the effect of winds and, for massive galaxies, also by AGN feedback, resulting in lower stellar masses than in the NF model.

Star Formation Rates

To confirm the previous findings the star formation rates are compared in Fig. 5.6. At very high redshifts z >4, the SFRs in the simulations are much higher than in the

SAMs. Only by assuming more efficient star formation in the NF SAM (τ∗ = 0.1) a

reasonable match to the simulations is obtained. However, at z < 1.5, the high SFE

model results in similar SFRs as the original NF model, as the larger SF efficiencies at high redshift lead to a more rapid depletion of the cold gas. In the simulation, the cold gas is rapidly turned into stars, resulting in lower SFRs at low redshifts compared to SAMs because of gas depletion. Only the FULL model shows a strongly decreasing SFR with decreasing redshift due to radio mode feedback, which becomes especially important for low redshifts and large halo masses. This result is consistent with the study of Saro et al.(2010), who compared their stripped-down versions of SAMs (with no feedback) to simulations, and found find higher SFRs in the simulations for all

Figure 5.6: Evolution of the star formation rates in simulations and SAMs for three mass bins. The mass bins and colors are the same as in Fig. 5.2. At high redshifts, SFRs are higher in the simulations than in the SAMs leading to more rapid depletion of the cold gas with very low SFRs at low redshifts.

galaxies within a cluster (central and satellites) at high redshifts and lower SFRs at low redshifts. In addition, Stringer et al. (2010) find a similar discrepancy for the specific star formation rates at high redshifts (larger in simulations than in their SAM) and good agreement for low redshifts.

To better understand this discrepancy between simulations and SAMs we take a closer look at the respective implementations of star formation. According to Springel & Hernquist (2003), stars in the simulations are formed locally out of cold gas with the star formation rate density proportional to the local three-dimensional density of gas to the power of 1.5, ρSF ∝ ρ1.5/t∗. The star formation timescale t∗ was set to

approximate the observed local Schmidt-Kennicutt relation (SK) for a simulation of a smooth, isolated disk-dominated galaxy set up to resemble the Milky Way. In the SAMs, the cold gas is assumed to settle into smooth exponential disks and stars form according to the SK-relation, implemented in terms of surface densities.

In Fig. 5.7, the SFR surface density versus the surface density of the cold gas is plotted for the simulated galaxies at z = 2 and z = 4 within 1/10 rvir for all re-simulations. The black dashed line is the SK-relation assuming a Salpeter IMF (Salpeter, 1955), as given in the original Kennicutt papers and as implemented in the simulations following Springel & Hernquist (2003). We would naively expect the simulations to follow this line. The red solid line shows the SK-relation for a Chabrier IMF (Chabrier, 2003), as assumed in the SAMs. At a given gas surface density, the SFR surface densities of the simulations lie mostly above the expected SK-relation. The change of normalization associated with converting from Salpeter to Chabrier cannot account for the increased star formation efficiency in the simulations. In general, star formation in the cosmological simulations is about a factor of five more efficient than for simulations of smooth isolated disks using the identical model (see Springel & Hernquist, 2003). This discrepancy is a consequence of the clumpy structure of

cold gas in the cosmological simulations. In the clumps the gas can reach higher local densities than in the idealized smooth disks that have been used by Springel & Hernquist(2003) to calibrate the star formation timescale by matching the SK-relation. As the implemented SK-relation is not linear, the structure of the cold gas distribution plays an important role for the overall star formation efficiency within the galaxies (see

Teyssier et al. (2010a) for a discussion on galaxy mergers). In other words, for any star formation model with a non-linear dependence on the local gas density (exponent larger than unity), a more clumpy gas distribution will effectively increase the star formation efficiency. These combined effects explain the much higher SF efficiencies at high redshift in the simulations relative to the SAMs.

Modes of Stellar Mass Growth

In the hierarchical picture, galaxies can grow their stellar masses in two ways: 1) by converting cold gas into stars in-situ and 2) by accreting already formed stars via

Figure 5.7: Star formation rate surface densities versus cold gas surface densities for the simulated galaxies within 1/10rvir. Black and blue stars correspond to different re-

simulations at z = 2or z = 4, respectively. The red solid line illustrates the Kennicutt- relation implemented in the SAM assuming a Chabrier-IMF, the black dashed line the one in the simulations (Springel & Hernquist, 2003) assuming a Salpeter-IMF.

mergers. These two modes are referred to as “in-situ” and “accreted”. The simulations exhibit two phases of growth, with a rapid early phase atz >2 during which stars are

formed in-situ from infalling cold gas, followed by an extended phase at z <3 during

which the growth is primarily due to accretion of stars formed in external galaxies (Oser et al.,2010). It is now investigated whether the SAMs show the same behavior. Fig. 5.8shows the fraction of cumulative in-situ over accreted stellar mass as a function of redshift for the three different mass bins. For the SAMs the qualitative trend of a decreasing fraction of in-situ growth is reproduced for the high mass bin. However, the fraction of in-situ formed stars dominates over accreted stars for all models, all masses, and at all redshifts. This is in contrast with the simulations, where accretion dominates over in-situ formation for massive systems at low redshifts as discussed in

Oser et al. (2010).

I note several interesting trends in the in-situ to accreted fraction as vary the physics in the SAMs. Adding thermal SN feedback increases the in-situ fraction at all redshifts and in all mass bins. This is presumably because it suppresses star formation in low- mass satellites which are the source of accreted stars. Adding the SN-driven winds and metal cooling further increases the in-situ fraction, again at all redshifts below z _∼4.

Switching on AGN feedback increases the in-situ fraction at high redshift and decreases it at low redshift in the high mass bin (and to a lesser extent in the intermediate mass bin). This is because the radio mode feedback shuts off cooling at late times in massive

Figure 5.8: Fraction of in-situ and accreted stellar mass versus redshift. The mass bins and colors are the same as in Fig. 5.2. The bimodal behavior as seen in simulations cannot be reproduced by the SAMs. Here, the in-situ star formation is dominating over accretion at all redshifts.

Figure 5.9: Comparison of the mean in-situ (left column) and accreted (right column) stellar masses. The mass bins and colors are the same as in Fig. 5.2. The in-situ stellar masses in the SAMs agree with the ones in the simulations reasonably well, whereas in the SAMs the accreted stellar mass is smaller than in the simulations.

halos, removing the supply of new gas needed to fuel ongoing in-situ star formation. Interestingly, increasing the star formation efficiency in the NF model has almost no effect on the in-situ to accreted fraction. This is presumably because the SF efficiency is increased in the central and (accreted) satellite galaxies alike. However, if the SFE were higher in high redshift galaxies than at low redshift, this would presumably in- crease the accreted fraction in present day galaxies. This may be part of the reason for the higher accreted fractions in the simulations.

Fig. 5.9 shows the evolution of the cumulative mass of in-situ and accreted stars separately for the simulations and the various SAM variants. Here one can see that the NF SAM actually reproduces the growth of in-situ stellar masses fairly well, though

overproducing the in-situ mass at low redshift somewhat, especially in the highest mass bin. One can speculate that gravitational heating in the simulations may prevent some of the late cooling in the highest mass bin and leads to lower in-situ stellar mass than the NF SAM (Naab et al., 2007;Johansson et al.,2009b; Feldmann et al.,2010). The radio mode AGN feedback in the FULL model leads to a similar suppression of this in-situ mass growth in the massive halos. The NF model with increased SFE gives an even better match to the simulations at high redshift. The discrepancy arises from the much lower accreted masses in the SAM. Here again, the SF model with high SFE comes the closest to matching the simulation results, but it still falls short by a con- siderable amount.

Part of the reason for the lower predicted accreted masses in the SAMs is that the SAMs used here only allow for cooling onto the central galaxy in the halo, effectively assuming that the hot gas reservoir of a satellite galaxy is stripped as soon as it enters the virial radius of the host. This is known to result in satellites that are too red and have star formation rates that are too low compared with observations (Kimm et al.,

2009). It will also truncate their star formation, resulting in a smaller amount of stellar mass that will eventually be accreted when they merge (Khochfar & Ostriker, 2008).